The Lefschetz properties are desirable properties of graded artinian algebras, which have strong consequences on the structure of their Hilbert functions. Their investigation is similar in flavor to what one does in algebraic geometry when studying the generic hyperplane section of a projective variety. The talk will provide a gentle introduction to this area of research and will illustrate the beautiful connections which occur when the above mentioned properties are translated into the language of combinatorics, algebraic geometry and representation theory. Some of these connections are quite surprising and still not completely understood.